TY - JOUR

T1 - Type II blow-up mechanisms in a semilinear heat equation with critical Joseph–Lundgren exponent

AU - Seki, Yukihiro

N1 - Funding Information:
The author was partly supported by Grant-in-Aid for Research Activity Start-up (No. 6887027 ) and Kyushu University Interdisciplinary Programs in Education and Projects in Research Development (No. 2012DB4000 ).
Publisher Copyright:
© 2018 Elsevier Inc.

PY - 2018/12/15

Y1 - 2018/12/15

N2 - We are concerned with blow-up mechanisms in a semilinear heat equation ut=Δu+|u|p−1u,x∈RN,t>0, where p>1 is a constant. It is well known that type II blow-up does occur if N≥11 and p>pJL, where pJL stands for the Joseph–Lundgren exponent: pJL={+∞,N≤10,1+[Formula presented],N≥11. On the other hand, it is also known that, if pJL, type II blow-up cannot occur under mild assumptions on initial data as far as nonnegative radially symmetric solutions are concerned. It has long remained open whether or not type II blow-up occurs for the borderline case p=pJL,N≥11. In the present article we prove constructively the existence of type II blow-up solutions for the Joseph–Lundgren critical case, thus giving an answer to this question.

AB - We are concerned with blow-up mechanisms in a semilinear heat equation ut=Δu+|u|p−1u,x∈RN,t>0, where p>1 is a constant. It is well known that type II blow-up does occur if N≥11 and p>pJL, where pJL stands for the Joseph–Lundgren exponent: pJL={+∞,N≤10,1+[Formula presented],N≥11. On the other hand, it is also known that, if pJL, type II blow-up cannot occur under mild assumptions on initial data as far as nonnegative radially symmetric solutions are concerned. It has long remained open whether or not type II blow-up occurs for the borderline case p=pJL,N≥11. In the present article we prove constructively the existence of type II blow-up solutions for the Joseph–Lundgren critical case, thus giving an answer to this question.

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U2 - 10.1016/j.jfa.2018.05.008

DO - 10.1016/j.jfa.2018.05.008

M3 - Article

AN - SCOPUS:85048471748

SN - 0022-1236

VL - 275

SP - 3380

EP - 3456

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 12

ER -